# Properties

 Label 975.1.bi.a Level $975$ Weight $1$ Character orbit 975.bi Analytic conductor $0.487$ Analytic rank $0$ Dimension $16$ Projective image $D_{20}$ CM discriminant -39 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 975.bi (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.486588387317$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{40})$$ Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{20}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{2} -\zeta_{40}^{14} q^{3} + ( \zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{4} -\zeta_{40} q^{5} + ( \zeta_{40}^{3} - \zeta_{40}^{9} ) q^{6} + ( -\zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{8} -\zeta_{40}^{8} q^{9} +O(q^{10})$$ $$q + ( \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{2} -\zeta_{40}^{14} q^{3} + ( \zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{4} -\zeta_{40} q^{5} + ( \zeta_{40}^{3} - \zeta_{40}^{9} ) q^{6} + ( -\zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{8} -\zeta_{40}^{8} q^{9} + ( -\zeta_{40}^{10} + \zeta_{40}^{16} ) q^{10} + ( \zeta_{40} - \zeta_{40}^{3} ) q^{11} + ( -\zeta_{40}^{4} + \zeta_{40}^{12} - \zeta_{40}^{18} ) q^{12} -\zeta_{40}^{18} q^{13} + \zeta_{40}^{15} q^{15} + ( -1 - \zeta_{40}^{2} + \zeta_{40}^{8} - \zeta_{40}^{14} - \zeta_{40}^{16} ) q^{16} + ( -\zeta_{40}^{3} - \zeta_{40}^{17} ) q^{18} + ( -\zeta_{40}^{5} + \zeta_{40}^{11} - \zeta_{40}^{19} ) q^{20} + ( \zeta_{40}^{10} - \zeta_{40}^{12} - \zeta_{40}^{16} + \zeta_{40}^{18} ) q^{22} + ( -\zeta_{40} + \zeta_{40}^{7} - \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{24} + \zeta_{40}^{2} q^{25} + ( \zeta_{40}^{7} - \zeta_{40}^{13} ) q^{26} -\zeta_{40}^{2} q^{27} + ( -\zeta_{40}^{4} + \zeta_{40}^{10} ) q^{30} + ( \zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{9} - \zeta_{40}^{11} + \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{32} + ( -\zeta_{40}^{15} + \zeta_{40}^{17} ) q^{33} + ( \zeta_{40}^{6} - \zeta_{40}^{12} + \zeta_{40}^{18} ) q^{36} -\zeta_{40}^{12} q^{39} + ( -1 + \zeta_{40}^{6} + \zeta_{40}^{8} - \zeta_{40}^{14} ) q^{40} + ( \zeta_{40}^{5} + \zeta_{40}^{11} ) q^{41} + ( \zeta_{40}^{8} + \zeta_{40}^{12} ) q^{43} + ( \zeta_{40} + \zeta_{40}^{5} - \zeta_{40}^{7} - \zeta_{40}^{11} + \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{44} + \zeta_{40}^{9} q^{45} + ( -\zeta_{40}^{11} - \zeta_{40}^{17} ) q^{47} + ( \zeta_{40}^{2} - \zeta_{40}^{8} - \zeta_{40}^{10} + \zeta_{40}^{14} + \zeta_{40}^{16} ) q^{48} - q^{49} + ( \zeta_{40}^{11} - \zeta_{40}^{17} ) q^{50} + ( \zeta_{40}^{2} - \zeta_{40}^{8} + \zeta_{40}^{16} ) q^{52} + ( -\zeta_{40}^{11} + \zeta_{40}^{17} ) q^{54} + ( -\zeta_{40}^{2} + \zeta_{40}^{4} ) q^{55} + ( \zeta_{40}^{7} + \zeta_{40}^{9} ) q^{59} + ( \zeta_{40}^{5} - \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{60} + ( -\zeta_{40}^{6} - \zeta_{40}^{18} ) q^{61} + ( 1 - \zeta_{40}^{4} - \zeta_{40}^{6} + \zeta_{40}^{10} + \zeta_{40}^{12} + \zeta_{40}^{14} - \zeta_{40}^{18} ) q^{64} + \zeta_{40}^{19} q^{65} + ( \zeta_{40}^{4} - \zeta_{40}^{6} - \zeta_{40}^{10} + \zeta_{40}^{12} ) q^{66} + ( \zeta_{40}^{13} - \zeta_{40}^{15} ) q^{71} + ( \zeta_{40} - \zeta_{40}^{7} + \zeta_{40}^{13} + \zeta_{40}^{15} ) q^{72} -\zeta_{40}^{16} q^{75} + ( \zeta_{40} - \zeta_{40}^{7} ) q^{78} + ( \zeta_{40} + \zeta_{40}^{3} - \zeta_{40}^{9} + \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{80} + \zeta_{40}^{16} q^{81} + ( \zeta_{40}^{6} + \zeta_{40}^{14} ) q^{82} + ( -\zeta_{40}^{13} + \zeta_{40}^{19} ) q^{83} + ( -\zeta_{40} + \zeta_{40}^{3} + \zeta_{40}^{7} + \zeta_{40}^{17} ) q^{86} + ( 1 - \zeta_{40}^{2} - \zeta_{40}^{6} + \zeta_{40}^{10} + \zeta_{40}^{14} - \zeta_{40}^{16} ) q^{88} + ( -\zeta_{40}^{7} - \zeta_{40}^{17} ) q^{89} + ( \zeta_{40}^{4} + \zeta_{40}^{18} ) q^{90} + ( 1 - \zeta_{40}^{12} ) q^{94} + ( -\zeta_{40}^{3} - \zeta_{40}^{5} + \zeta_{40}^{9} + \zeta_{40}^{11} - \zeta_{40}^{17} - \zeta_{40}^{19} ) q^{96} + ( -\zeta_{40}^{9} + \zeta_{40}^{15} ) q^{98} + ( -\zeta_{40}^{9} + \zeta_{40}^{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} + 4q^{9} + O(q^{10})$$ $$16q + 4q^{4} + 4q^{9} - 4q^{10} - 16q^{16} - 4q^{30} - 4q^{36} - 4q^{39} - 20q^{40} - 16q^{49} + 4q^{55} + 16q^{64} + 8q^{66} + 4q^{75} - 4q^{81} + 20q^{88} + 4q^{90} + 12q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{40}^{4}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
194.1
 −0.156434 − 0.987688i −0.987688 + 0.156434i 0.987688 − 0.156434i 0.156434 + 0.987688i 0.891007 + 0.453990i 0.453990 − 0.891007i −0.453990 + 0.891007i −0.891007 − 0.453990i 0.891007 − 0.453990i 0.453990 + 0.891007i −0.453990 − 0.891007i −0.891007 + 0.453990i −0.156434 + 0.987688i −0.987688 − 0.156434i 0.987688 + 0.156434i 0.156434 − 0.987688i
−1.69480 + 0.550672i −0.587785 0.809017i 1.76007 1.27877i 0.156434 + 0.987688i 1.44168 + 1.04744i 0 −1.23134 + 1.69480i −0.309017 + 0.951057i −0.809017 1.58779i
194.2 −0.863541 + 0.280582i 0.587785 + 0.809017i −0.142040 + 0.103198i 0.987688 0.156434i −0.734572 0.533698i 0 0.627399 0.863541i −0.309017 + 0.951057i −0.809017 + 0.412215i
194.3 0.863541 0.280582i 0.587785 + 0.809017i −0.142040 + 0.103198i −0.987688 + 0.156434i 0.734572 + 0.533698i 0 −0.627399 + 0.863541i −0.309017 + 0.951057i −0.809017 + 0.412215i
194.4 1.69480 0.550672i −0.587785 0.809017i 1.76007 1.27877i −0.156434 0.987688i −1.44168 1.04744i 0 1.23134 1.69480i −0.309017 + 0.951057i −0.809017 1.58779i
389.1 −1.16110 1.59811i −0.951057 0.309017i −0.896802 + 2.76007i −0.891007 0.453990i 0.610425 + 1.87869i 0 3.57349 1.16110i 0.809017 + 0.587785i 0.309017 + 1.95106i
389.2 −0.183900 0.253116i 0.951057 + 0.309017i 0.278768 0.857960i −0.453990 + 0.891007i −0.0966818 0.297556i 0 −0.565985 + 0.183900i 0.809017 + 0.587785i 0.309017 0.0489435i
389.3 0.183900 + 0.253116i 0.951057 + 0.309017i 0.278768 0.857960i 0.453990 0.891007i 0.0966818 + 0.297556i 0 0.565985 0.183900i 0.809017 + 0.587785i 0.309017 0.0489435i
389.4 1.16110 + 1.59811i −0.951057 0.309017i −0.896802 + 2.76007i 0.891007 + 0.453990i −0.610425 1.87869i 0 −3.57349 + 1.16110i 0.809017 + 0.587785i 0.309017 + 1.95106i
584.1 −1.16110 + 1.59811i −0.951057 + 0.309017i −0.896802 2.76007i −0.891007 + 0.453990i 0.610425 1.87869i 0 3.57349 + 1.16110i 0.809017 0.587785i 0.309017 1.95106i
584.2 −0.183900 + 0.253116i 0.951057 0.309017i 0.278768 + 0.857960i −0.453990 0.891007i −0.0966818 + 0.297556i 0 −0.565985 0.183900i 0.809017 0.587785i 0.309017 + 0.0489435i
584.3 0.183900 0.253116i 0.951057 0.309017i 0.278768 + 0.857960i 0.453990 + 0.891007i 0.0966818 0.297556i 0 0.565985 + 0.183900i 0.809017 0.587785i 0.309017 + 0.0489435i
584.4 1.16110 1.59811i −0.951057 + 0.309017i −0.896802 2.76007i 0.891007 0.453990i −0.610425 + 1.87869i 0 −3.57349 1.16110i 0.809017 0.587785i 0.309017 1.95106i
779.1 −1.69480 0.550672i −0.587785 + 0.809017i 1.76007 + 1.27877i 0.156434 0.987688i 1.44168 1.04744i 0 −1.23134 1.69480i −0.309017 0.951057i −0.809017 + 1.58779i
779.2 −0.863541 0.280582i 0.587785 0.809017i −0.142040 0.103198i 0.987688 + 0.156434i −0.734572 + 0.533698i 0 0.627399 + 0.863541i −0.309017 0.951057i −0.809017 0.412215i
779.3 0.863541 + 0.280582i 0.587785 0.809017i −0.142040 0.103198i −0.987688 0.156434i 0.734572 0.533698i 0 −0.627399 0.863541i −0.309017 0.951057i −0.809017 0.412215i
779.4 1.69480 + 0.550672i −0.587785 + 0.809017i 1.76007 + 1.27877i −0.156434 + 0.987688i −1.44168 + 1.04744i 0 1.23134 + 1.69480i −0.309017 0.951057i −0.809017 + 1.58779i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 779.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
13.b even 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner
325.p even 10 1 inner
975.bi odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.1.bi.a 16
3.b odd 2 1 inner 975.1.bi.a 16
13.b even 2 1 inner 975.1.bi.a 16
25.e even 10 1 inner 975.1.bi.a 16
39.d odd 2 1 CM 975.1.bi.a 16
75.h odd 10 1 inner 975.1.bi.a 16
325.p even 10 1 inner 975.1.bi.a 16
975.bi odd 10 1 inner 975.1.bi.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.1.bi.a 16 1.a even 1 1 trivial
975.1.bi.a 16 3.b odd 2 1 inner
975.1.bi.a 16 13.b even 2 1 inner
975.1.bi.a 16 25.e even 10 1 inner
975.1.bi.a 16 39.d odd 2 1 CM
975.1.bi.a 16 75.h odd 10 1 inner
975.1.bi.a 16 325.p even 10 1 inner
975.1.bi.a 16 975.bi odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(975, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 92 T^{4} - 228 T^{6} + 230 T^{8} - 72 T^{10} + 17 T^{12} - 4 T^{14} + T^{16}$$
$3$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$5$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16}$$
$13$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$1 - 4 T^{2} + 92 T^{4} + 228 T^{6} + 230 T^{8} + 72 T^{10} + 17 T^{12} + 4 T^{14} + T^{16}$$
$43$ $$( 5 + 5 T^{2} + T^{4} )^{4}$$
$47$ $$1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16}$$
$53$ $$T^{16}$$
$59$ $$1 + 16 T^{2} + 97 T^{4} - 32 T^{6} + 150 T^{8} + 32 T^{10} + 12 T^{12} + 4 T^{14} + T^{16}$$
$61$ $$( 25 + 25 T^{2} + 10 T^{4} + T^{8} )^{2}$$
$67$ $$T^{16}$$
$71$ $$1 - 4 T^{2} + 92 T^{4} + 228 T^{6} + 230 T^{8} + 72 T^{10} + 17 T^{12} + 4 T^{14} + T^{16}$$
$73$ $$T^{16}$$
$79$ $$T^{16}$$
$83$ $$1 - 16 T^{2} + 97 T^{4} + 32 T^{6} + 150 T^{8} - 32 T^{10} + 12 T^{12} - 4 T^{14} + T^{16}$$
$89$ $$( 16 + 8 T^{2} + 4 T^{4} + 2 T^{6} + T^{8} )^{2}$$
$97$ $$T^{16}$$